maximum principle

Let f:U→ℝ (where U⊆ℝd) be a harmonic function. Then f attains its extremal values on any compactK⊆U on the boundary ∂⁡K of K. If f attains an extremal value anywhere in the interior of K, then it is constant.

Maximal modulus principle

Let f:U→ℂ (where U⊆ℂ) be a holomorphic function. Then |f| attains its maximal value on any compact K⊆U on the boundary ∂⁡K of K. If |f| attains its maximal value anywhere on the interior of K, then it is constant.